The Stacks project

Proof. Let $T$ be a scheme over $B$ and let $Y \subset (Z \times _ B X)_ T$ be an element of $\mathrm{Hilb}_{Z \times _ B X/B}(T)$. Then we see that $Y$ is the graph of a morphism $Z_ T \to X_ T$ over $T$ if and only if $k = \text{pr}_1|_ Y : Y \to Z_ T$ is an isomorphism. By More on Morphisms of Spaces, Lemma 76.49.6 there exists an open subscheme $V \subset T$ such that for any morphism of schemes $T' \to T$ we have $k_{T'} : Y_{T'} \to Z_{T'}$ is an isomorphism if and only if $T' \to T$ factors through $V$. This proves the lemma. $\square$